A bit of history from Wikipedia:
"August Kundt, in 1866, demonstrated an acoustical standing wave by placing seeds of lycopodium or cork dust in a tube. When a sound was made in the tube, the material inside lined up in nodes and antinodes in line with the oscillation of the wave, creating a standing wave. Later that century, Behn showed that small flames could be used as sensitive indicators of pressure. Finally, in 1905, using these two important discoveries, Heinrich Rubens, whom this experiment is named after, took a 4-metre-long tube and drilled 200 small holes into it at 2 centimeter intervals, and filled it with a flammable gas. After lighting the gas (whose flames all rose to near-equal heights), he noted that a sound produced at one end of the tube would create a standing wave, equivalent to the wavelength of the sound being made."
In many basic physics classes the Ruben's tube is used as a demonstration of the relationship between frequency and wavelength. Students visually measure standing wave. Then, because the speed of sound in propane is 235 m/s, the students can calculate the frequency being played into the Ruben's tube.
As the sine waves are played through the propane, they compress the propane in certain sections thereby increasing its density. This in turn increases the pressure, and because the area is at a high pressure (and air likes to flow from high to low pressure... thank you Le Chatelier) more air molecules are forced out these holes in an effort to reach equilibrium causing the flames rise higher. In areas where the propane is not as compressed, the area specific pressure is lower resulting in lower flame heights. Essentially when you are visualizing the wave with fire you are seeing a standing wave.
Since we know that frequency (f) equals speed (v) divided by wavelength (lamba) we are able to quantitatively determine the frequencies running through the Rubens tube. (f=v/lambda) Now, when doing this, remember that "v" is not the velocity of sound in air, (roughly 344 m/s) but rather the velocity of sound in your gas ( in this case propane, so 235 m/s). Also keep in mind that elevation and temperature affect this, but 235 is a good estimate. So, measure your wavelength from peak to peak (flame maxima) or trough to trough (flame minima), plug into the equation, and you get your frequency.
After some digging at the college library, I was able to unearth the original journal publication by von Heinrich Rubens:
It was published on December, 8th 1904 in the Verhandlungen der Deutschen Physikalischen Gesellschaft- a German physics journal. The original tube used was brass. It was four meters long, 8 centimeters wide, and had about 100 holes which were each about 2 mm wide. He used Coal Gas which was inserted on one end through a brass cap, and then covered the other end with a rubber diaphragm. Interestingly his sources of sound were bells, an organ pipe, or tuning forks.